Local saturation of the non-stationary ideal over P_κ λ

Starting with a λ-supercompact cardinal κ, where λ is a regular cardinal greater than or equal to κ, we produce a model with a stationary subset S of P_κ λ such that NS_{κ λ} | S, the ideal generated by the non-stationary ideal NS_{κ λ} over P_κ λ together with P_κ λ {set minus} S, is λ⁺-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). We also show that in our model we can make NS_{κ λ} | S (κ, λ) λ⁺-saturated, where S (κ, λ) is the set of all x ∈ P_κ λ such that ot (x), the order type of x, is a regular cardinal and x is stationary in sup (x). Furthermore we construct a model where NS_{κ λ} | S (κ, λ) is κ⁺-saturated but GCH fails. We show that if S {set minus} S (κ, λ) is stationary in P_κ λ, then S can be split into λ many disjoint stationary subsets.